Functions, Domain, and Range

Functions

Identifying Functions

• Vertical Line Test: A graph represents a function if any vertical line drawn intersects the curve at no more than one point.

  • If a vertical line intersects the graph at more than one point, the graph does not represent a function.

  • Example:

    • A curve $y = f(x)$ where any vertical line intersects at only one point is a function.
    • A curve where a vertical line intersects at multiple points is not a function.

Definition of a Function

• A function is defined as a set of ordered pairs.

Ordered Pairs

• Given sets A and B, the Cartesian product A × B is a set of ordered pairs.
  • Example: If $A = {1, 2}$ and $B = {2, 3}$, then $A \times B = {(1, 2), (1, 3), (2, 2), (2, 3)}$.
• Each member of the set A × B is an ordered pair (e.g., (1, 2), (1, 3)).

Relation

• Any set of ordered pairs is called a relation.
  • A × B is a relation.

Function (Formal Definition)

• A function is a special type of relation in which no two distinct elements (ordered pairs) have the same first entry.
  • If $(a, b)$ is an element, a is the first entry, and b is the second entry.
• In a function, if two ordered pairs have the same first entry, their second entries must also be the same.

Examples

• Example A: $f = {(1, 2), (-1, 2), (3, 4), (2, 6)}$ is a function because all first entries are distinct.
• Example B: $g = {(2, -1), (3, 5), (2, 4)}$ is NOT a function because the first entry 2 is repeated with different second entries (-1 and 4).
  • For g to be a function, If the first entries are the same, the second entries must also be the same.
• Example C: $h = {(a, d), (c, d), (b, d)}$ is a function because the first entries (a, c, b) are distinct.

Domain and Range

Domain

• The domain of a function is the collection of all first entries (x-values) in the ordered pairs.
  • $D = {x : y = f(x)\text{ is defined}}$
  • Analogy: The domain represents the valid inputs into a CPU (Central Processing Unit), where the output is a real number.
  • If an input results an undefined or error output, that input is not in the domain.

Example Domain

• For $f = {(1, 2), (-1, 2), (3, 4), (2, 6)}$, the domain is $D_f = {1, -1, 3, 2}$.

Range

• The range of a function is the collection of all second entries (y-values) in the ordered pairs.
  • For the same example $f = {(1, 2), (-1, 2), (3, 4), (2, 6)}$ the range is $R_f = {2, 4, 6}$.
    • Note: Elements in a set are not repeated.

Mathematical Definition of Domain and Range

• Let $y = f(x)$ be a function.
  • The domain (D) of f is the collection of x-values such that $f(x)$ is defined (a real number).
  • The range (R) is given by the y-values (f(x)) for x in D.

Calculator Analogy

• If a calculator returns "error," the input is not in the domain.

Determining Domain Algebraically

Example 1

• Consider $f(x) = \frac{1}{x - 1}$.
  • If $x = 0$, $f(0) = \frac{1}{0 - 1} = -1$ (real number), so 0 is in the domain.
  • If $x = 1$, $f(1) = \frac{1}{1 - 1} = \frac{1}{0}$ (undefined), so 1 is not in the domain.
  • The domain of $f(x)$ is all real numbers except $x = 1$.
• In interval notation: $(-\infty, 1) \cup (1, \infty)$.

Example 2

• Consider $g(x) = \sqrt{x - 2}$.
  • If $x = -1$, $g(-1) = \sqrt{-1 - 2} = \sqrt{-3}$ (not real), so -1 is not in the domain.
  • If $x = 0$, $g(0) = \sqrt{0 - 2} = \sqrt{-2}$ (not real), so 0 is not in the domain.
• To find the domain, set the expression inside the square root to greater than or equal to zero:
  • $x - 2 \geq 0$
  • $x \geq 2$
  • The domain of $g(x)$ is all real numbers greater than or equal to 2.
• In interval notation: $[2, \infty)$.
• Graphing the function shows that x must be at least 2, with y increasing from 0.
• For a function of the form $sqrt{x-a}$, the function starts at the the x value of 'a' and continues to increase.

Range of Function

• Let f be a function with domain D. Then the range R is given by the y values f(x) such that $x \in D$

Example (Range)

• For $g(x) = \sqrt{x - 2}$, the domain is x >= 2.
  • To sketch the graph, take x-values from 2 and above.
  • If x = 2, g(2) = 0.
  • If x = 3, g(3) = 1.
  • If x = 4, g(4) = \sqrt{2}.
  • The y-values are increasing, with the least value being 0.
  • The range is from 0 to infinity: $[0, \infty)$.

Additional example and discussion

Example 3

• Consider $h(x) = \frac{1}{\sqrt{x - 2}}$.
• To find the domain, set the expression inside the square root to zero.
  • $\sqrt{x - 2} = 0$
  • $x - 2 = 0$
  • $x = 2$
• If x < 2, the output is not real.
• There isn't a hole at 2 because the formula does not exist.
• In interval notation: $(2, \infty)$.
• To find the range, find all possible output values in interval $(2, \infty)$
• As x gets very large, the denominator also gets very large too.
  Thus, the reciprocal would get very small getting closer to 0.
• Thus, the range of all posible y values is in the form (0,1].